Maximal-coefficient conjecture for Stern polynomials

Let

$$B_{n}(t)=\sum_{i=0}^{e(n)}a_{i,e(n)}x^{i}$$

and define

$$\mathcal{M}(n)=\operatorname{max}\{a_{i,e(n)}:\;i=1,2,\ldots,e(n)\}.$$

Then the following equality holds:

$$\operatorname{max}\{\mathcal{M}(m):\;m\in[2^{n-1},2^{n}]\}=\operatorname{max}\{C(n,0),C(n-1,1),\ldots,C(n-k,k)\},$$

where $k$ is equal to $n/2$ if $n$ is even and $(n-1)/2$ if $n$ is odd.

This conjecture arose from investigations of the sequence of maximal coefficients of the Stern polynomial $B_n(t)$. The supplied context does not define the coefficient notation $a_{i,e(n)}$ or the quantities $C(n,j)$, so those definitions and the conjecture's current status should be checked against the source.