What is the name of the word problem for free groups under straight line program encoding?
The blog post you link to already gives a deterministic polynomial-time (in fact, linear-time) algorithm for the word problem over a free group with 2 letters.

In contrast, no deterministic polynomial-time algorithm for identity testing for polynomials is currently known (this is a famous open problem).

Therefore, it's not likely to be easy to prove that the word problem over the free group is equivalent in complexity to identity testing of polynomials.



The "straight-line encoding" doesn't seem to change the complexity of the word problem in any interesting way.  It seems equivalent to simply specifying the input as an expression (i.e., a sequence of symbols, where each symbol is either $x_i$ or $x_i^{-1}$).