I've been following this discussion with some interest, and must say I believe I needn't worry about either workflow option if I always start from from my raw file, use Lightroom as my primary photo editor and repurpose the raw files as needed, be it for devices, or for any particular printer/paper combination. The algorithms within LR for doing these on-the-fly conversions seem to be producing coherent results, taking into account that for printing purposes one should be making final luminance and colour adjustments under soft-proof bespoke to the printer/paper combination in play. So, that's the practical aspect of interest to me. Also of interest would be to see an explanation of the principles underlying the added workflow conversion step Doug is recommending - conversion to the printer profile - before conversion to a smaller colour space. Of course each printer/paper combination has its own gamut space and all of them are smaller than ProPhoto. So the process Doug recommends essentially compresses the colour space for the image in two steps rather than one: from Pro-Photo to the Printer Profile space and from that space to sRGB. Why in principle is this better than a direct conversion from ProPhoto to sRGB? Or is it a result of empirical observation and the reason is as yet unclear?
(Typo corrected)
Here's the actual conversions done that demonstrates why the two step process is better when converting an image from ProPhoto to sRGB when trying to match a print made directly from Andrew's Printer Gamt Test File.
Focusing in on the Green, vertical gradient column locate a point equidistant from the bottom and left/right sides (a good location to take a verifying reading with a spectro) the ProPhoto RGB values are (7,255,7). This is, of course, an unrealizable color that can be thought of as an intense, supersaturated green. This is not even close to being reproducible in sRGB, aRGB, a print gamut, or anything in our reality.
Let's follow the two paths of color conversion.
The first is straightforward. Convert the color to sRGB then print it using either Perceptual or Relative colorimetry. Let's use Relative for simplicity noting that we could instead use Perceptual. The results in sRGB are (0,255,0) over the entire bottom third of the vertical green column. So the conversion results in clipping. Let's look at the details. PP(7,255,7) is, in XYZ (PCS working space for matrix profiles) is 13.6, 71.2, .128. Running the matrix conversion to sRGB and scaling to sRGB's almost 2.2 gamma yields sRGB(-221, 279, -108). Since us mortals, and Adobe, can't deal with RGB values beyond [0:255] they are simply clipped to sRGB(0,255,0). That's a big haircut. But does that get us close to the printer's gamut? No, it does not. Virtually all colors near sRGB's three primaries cannot be printed. Printer gamuts exceed sRGB and aRGB only along the gamut triangle's sides.
So what happens when we print sRGB(0,255,0) and compare that to printing the ppRGB(7,255,7). First the sRGB(0,255,0) is converted to PCS (Lab) then the Lab values, adapted for BPC, use the 3D LUTs to interpolate a set of RGB values to send to the printer. This process is a function of the printer profile so YMWV. Often, printer profiles simply map out of gamut colors to the closest color on the gamut boundary.
Using my printer and profiles (a custom profile with Epson PPG printed on a Canon 9500 II) RC w BPC resulted in a Lab value of 72,-53, 67 as read by a spectro.
Now let's examine what happens when the image is printed from ProPhoto directly to the printer using RC w BPC. First the ppRGB(7,255,7) is converted to PCS (Lab clipped [-128:127]) then the BPC adapted 3D lookup is output to the printer and the image is printed. On my printer this produces a Lab value of (60, -67, 56) which is very far off (Deltas E:21) from the yellowish green printed by converting first to sRGB.
By going through the convert to printer then convert back to sRGB allows one to more closely match the printer's actual image as printed from RAW or ProPhoto RGB to an image intended for the web.
The problem's encountered in this is loss of accuracy due to inaccuracies in the printer's BtoA->AtoB. Each transition can be expected to introduce error ranging from tenths of a DeltaE to several DeltaEs at the gamut boundary. So, unless an image is significantly out of aRGB or sRGB, I don't recommend this. However, it's great for images with a lot of synthetic colors such as rainbow or the grainger color gradients. It's not accurate per se since the colors are out of gamut, but it does make for a much closer match between a print image and an image in sRGB intended for the web.