Cyclotomic factorization conjecture for tatami tiling generating polynomials

Let $T(n,z)$ be the generating polynomial for tatami tilings of an $n\times n$ square with $n$ monomers, where the exponent of $z$ records the number of vertical dimers. Let $S_k(z)=\prod_{i=1}^k(1+z^i)$ be the generating polynomial for subsets of $\{1,\ldots,k\}$ by their element sum, and let $P(n,z)$ be a polynomial.

Cyclotomic factorization conjecture. The generating polynomial $T(n,z)$ has the factorization

$$T(n,z)=P(n,z)\prod_{j\geq 1}S_{\left\lfloor\frac{n-1}{2^j}\right\rfloor}(z),$$

where $P(n,z)$ is an irreducible polynomial. The factorization is motivated by the correspondence between tatami tilings and subsets with prescribed sums, but the source gives no resolution of the conjecture.