Anthony's conjecture on the number of polynomial threshold functions

Let $T(n,d)$ be the number of $n$-variable polynomial threshold functions of degree $d$, and let

$$m=\binom{n}{\le d}.$$

Here $\binom{n}{\le d}$ denotes the number of monomials in at most $d$ variables’ degree contribution, including the constant term. Anthony's conjecture. For all degrees $1\le d\le n$, as $n\to\infty$,

$$T(n,d)=\left(2-o(1)\right)\binom{2^n-1}{\le m-1}.$$

The conjecture would make the paper's upper bound asymptotically sharp when the degree grows rapidly, including degrees linear in $n$; its resolution is not given in the supplied text.