Monotonicity conjecture for the asymptotic code rate

Let $q>2$ and let $\theta=1-1/q$. For $x$ in the relevant relative-distance range, write $\alpha(x)=\lim_{n\to\infty}n^{-1}\log_q A_q(n,xn)$, where $A_q(n,d)$ is the maximum size of a $q$-ary code of length $n$ and minimum Hamming distance at least $d$. Monotonicity conjecture. The function

$$\frac{\alpha(x)}{\theta-x}$$

is decreasing. Equivalently,

$$\alpha(tx+(1-t)\theta)\leq t\alpha(x)+(1-t)\alpha(\theta),\qquad t\in[0,1].$$

The conjecture is proposed as a weaker alternative to the unproved assertion that $\alpha(x)$ is cup-convex. The preceding hybrid Elias–Plotkin bound has the same relevant convexity and differentiability features, but it is not known whether the asserted monotonicity holds for the asymptotic rate function itself.