Simplicity conjecture for the characteristic polynomial of a randomized subtraction game

Let $t$ be the number of allowed moves, and let $d_1,\ldots,d_t$ be the corresponding positive integers, with characteristic polynomial

$$\chi_S(x)=tx^{d_t}+\sum_{i=1}^{t-1}x^{d_i}+1.$$

Simplicity conjecture. The roots of $\chi_S(x)$ are all simple whenever $d_t\equiv 1\pmod{2}$ and $d_i\equiv 0\pmod{2}$ for all $1\leq i\leq t-1$.

This conjecture concerns the case not covered by the preceding convergence theorem and would permit the authors' argument to proceed under the assumption that all roots are simple. The source gives no proof or resolution of the claim.