Conjecture on low-dimensional irreducible characters of the symmetric group

Let $k\geq 2$ be fixed and let $n$ be sufficiently large. Write $\lambda=[n-a,\lambda_2,\ldots,\lambda_t]\vdash n$, with $t<n-a$, and let $f^\lambda$ denote the dimension of the irreducible character of $\operatorname{Sym}(n)$ afforded by $\lambda$.

Low-dimensional character conjecture. If

$$f^\lambda<\binom{n}{k},$$

then $a\leq k-1$ or

$$\lambda\in\\{[n-k,k],[n-k,1^k]\\}.$$

This conjecture describes the possible partitions indexing irreducible characters of $\operatorname{Sym}(n)$ whose dimensions are below the threshold $\binom{n}{k}$, for fixed $k$ and sufficiently large $n$. The supplied text does not indicate whether the conjecture has been proved or disproved.