The strengthened monotonicity conjecture for multiplicities in $W_{n,n+K}$

Let $K\in\mathbb{N}$, let $\lambda\vdash 4K$, and write $\lambda^{(n-3K)}$ for the partition obtained by adding a first row of length $n-3K$ to $\lambda$. Let $m_{n,n+K}^{\lambda^{(n-3K)}}$ denote the multiplicity of the irreducible representation $S^{\lambda^{(n-3K)}}$ in $W_{n,n+K}$. Strengthened monotonicity conjecture. In Theorem, one can take $N_K=1$ and any $n\geq 3K$; equivalently, for every $K\in\mathbb{N}$, every partition $\lambda\vdash 4K$, and every $n\geq 3K$, the multiplicities form the asserted generally increasing sequence from this bound. The conjecture strengthens the preceding theorem, which establishes eventual general increase but does not identify the universal starting point. Its resolution would give uniform monotonicity beginning at the smallest proposed value of $N_K$.