Li's unimodality conjecture for bounded-composition largest parts

Let $a(k,n,q)$ denote the number of compositions of $k$ with $n$ parts whose largest part is $q$, and let $\binom{n,q}{k}$ denote the coefficient of $x^k$ in $(1+x+x^2+\cdots+x^{q-1})^n$. Let $c$ be a positive integer. Li's unimodality conjecture. For every $n$, the function

$$a((c+1)n,n,q)=\binom{n,q}{cn}-\binom{n,q-1}{cn}$$

is unimodal in $q$, with its maximum attained at $q=\lfloor\log_{1+\frac{1}{c}}n\rfloor+1$ or $q=\lfloor\log_{1+\frac{1}{c}}n\rfloor+1$. In particular, $a(2n,n,q)=\binom{n,q}{n}-\binom{n,q-1}{n}$ is unimodal in $q$, with maximum at $q=\lfloor\log_2 n\rfloor$ or $q=\lfloor\log_2 n\rfloor+1$. The conjecture concerns the distribution of compositions by their largest part; the stated maximizing values arise from the asymptotic scale of the bounded polynomial coefficients. The paper gives experimental motivation and later discusses the need for a combinatorial proof, but provides no resolution.