The finite-dimensional limiting dimension conjecture for finitely strictly singular adjustment

Let $X$ be a Banach space, and let $(M_n)_{\mathbb{N}'}$ and $(P_n)_{\mathbb{N}'}$ be sequences of subspaces of $X$, with $(P_n)_{\mathbb{N}'}$ finite-dimensional. For $\lambda<1$, suppose that $(M_n)_{\mathbb{N}'}$ is finitely strictly singular $\lambda$-adjusted with $(P_n)_{\mathbb{N}'}$. Finite-dimensional limiting dimension conjecture. There exists a constant integer $C$ such that

$$\dim M_n<\dim P_n+C$$

for all sufficiently large $n$. This is posed as an open question because the corresponding assertion for finitely strictly singular $\lambda$-adjustment is unproved in general, although it is known for Hilbert spaces and the analogous non-finitely-strictly-singular result is available under stronger assumptions.