The generalized Loewy length monotonicity conjecture under flat local extensions

Let $R\to S$ be a flat local extension between Cohen--Macaulay local rings with infinite residue fields. Write $g\ell\ell_R(R)$ and $g\ell\ell_S(S)$ for their generalized Loewy lengths. The generalized Loewy length monotonicity conjecture. If $R\to S$ is such an extension, then

$$g\ell\ell_R(R)\leqslant g\ell\ell_S(S).$$

This conjecture is motivated by Lech's conjecture on Hilbert--Samuel multiplicities and by known results on generalized Loewy length. The source describes it as a conjectural inequality and supplies no resolution, so its status remains open.