The short answer is "no," a black Arkansas provides considerably more polish.
However, comparing natural stones in general and Arkansas stones in particular to the grit screen size of synthetic stones is difficult and in some cases controversial.
Most synthetic stones are made with very consistent and carefull screened abrasives -- which in the case of waterstones all break down at the same time, and in the case of oilstones wear fairly evenly. Natural stones, as you'd expect, aren't as uniform.
Arkansas stones are even more peculiar because the abrasive material -- called Novaculite -- is one size and one size only no matter whether it's in a medium, fine, or extra-fine stone. The distinctions in performance aren't grit size but the "binding" substrates and the relative levels of Novaculite concentration. As if that weren't enough to make comparisons difficult, the most widely published and accepted conversion chart (originally from Norton/St. Gobain) underrates how fine Arkansas stones cut by quite a bit. At least in my experience as well as in that of nearly everyone else who uses Arks critically.
Let me add that part of the appeal of natural stones is that their polish is not perfectly even and a natural edge tends to be more slippery than a synthetic, but at the expense of some shine. It's easy to waste a lot of money on natural stones, so if you're interested it's important to get guidance from someone who knows the ropes. That goes triple for stones from Asia.
A Forschner can usefully be polished up to the 4K - 6K# (Japanese) range. I put a good "surgical black," like a Hall's, at a skosh finer than 4K and a really good translucent (lots of luck finding one) at around 5K. FWIW, there are some interesting European natural stones in the general grit range, like the "Belgian Blue."
Compared to other Arkansas stones, your 1.5K synthetic falls somewhere between "Soft" and "Hard." It depends on the particular stones.
There are several very good Japanese synthetic waterstones in the 4K to 6K "medium-fine" (as Japanese stones go) range. My favorite for price. speed and performance is the Arashiyama -- also called Takenoko -- it's/they're sometimes listed at 8,000 but 6,000 is right, trust me. The King 6000 is also a very good stone, but not as fast nor as fine as the Arashiyama, and needs more frequent flattening.
Hope this helps,
BDL