Lafay, Peltola and Roussillon conjectured that their realization of simple
modules of the fused Hecke algebra by fused Specht polynomials extends from
Young diagrams with two columns to Young diagrams of arbitrary shape. We
give a counterexample for
\[
n=8,\qquad \lambda=(3,2,2,1),\qquad
\varsigma=(2,2,2,2).
\]
More precisely, we exhibit an explicit linear dependence among three fused
Specht polynomials indexed by row-strict Young tableaux. The same example
also yields an infinite family of counterexamples.