The conjectural form of the Poincaré–Betti series for $A_n$

Let $A_n$ be the naturally graded $K$-algebra considered in the paper, let $r(n)$ be its number of variables, and let $C_{n,1}$ denote the stated coefficient, with $C_{n,1}=\lceil n/2\rceil$. For the Poincaré–Betti series, write

$$P(\operatorname{Tor}_{*}^{A_n}(K,K),t)=-\frac{(1+t)^{\ell_1(n)}}{q_n(t)},\qquad q_n(t)=\sum_{i=0}^{\ell_2(n)}h_i(n)t^i.$$

Conjectural Poincaré–Betti series formula. The polynomial $q_n(t)$ satisfies $q_n(-1)\neq 0$, and

$$\ell_1(n)=\#\{\text{odd primes }p:p^2\leq n\},\qquad \ell_2(n)=\ell_1(n)+1,$$

with

$$h_0(n)=-1,\qquad h_1(n)=r(n)-\ell_1(n),\qquad h_{\ell_2(n)}(n)=C_{n,1}=\lceil n/2\rceil.$$

This proposes a precise numerator and denominator structure for the Poincaré–Betti series of the algebras $A_n$. The supplied text does not indicate whether the assertion is proved or remains open.