Matsaev's conjecture for contractions on commutative $L^p$-spaces

Let $1<p<\infty$ with $p\ne 2$, let $\Omega$ be a measure space, and let $T$ be a contraction on $L^p(\Omega)$. Let $S$ be the right shift on $\ell^p$, defined by

$$S(a_0,a_1,a_2,\ldots)=(0,a_0,a_1,a_2,\ldots).$$

For a complex polynomial $P$, write $P(T)$ and $P(S)$ for the corresponding polynomial operators. Matsaev's conjecture. One has

$$\left\|P(T)\right\|_{L^p(\Omega)\to L^p(\Omega)}{\leqslant} \left\|P(S)\right\|_{\ell^p\to\ell^p}.$$

This is the classical commutative form of Matsaev's conjecture, introduced in 1971 and motivating its noncommutative analogue. The validity of the inequality is known for certain classes of contractions, while for all other values of $p$ it remains open for arbitrary contractions.