A degree-linear upper bound for the Pólya exponent of forms with fixed arity

Let $P = \sum_ω a_ωx^ω$ be a degree $d$ homogeneous polynomial in $x_0, \dots, x_n$ with real coefficients. Pólya showed that, if $P$ is strictly positive on the standard $n$-simplex, then $(x_0 + \cdots + x_n)^M P$ has strictly positive coefficients for all $M \gg 0$. The least such $M$ is known as