Pattern for H2 of SL2 over truncated polynomial rings over F2

Determine whether, for all integers n ≥ 2, the Schur multiplier H_2(SL_2(F_2[X]/(X^n)), Z) is isomorphic to (Z/2)^n, extending the observed pattern for n=2,…,5.

Background

Using GAP computations, the authors observe that for small n (2 ≤ n ≤ 5), the second homology group of SL_2 over the truncated polynomial ring F_2[X]/(X^n) appears to be a direct sum of n copies of Z/2.

They explicitly raise the question of whether this pattern persists for all n ≥ 2, which would provide a uniform description of the Schur multiplier in this family and inform broader structural understanding of homology for linear groups over nilpotent extensions.