Log-concavity of annular Khovanov homology

Annular Khovanov homology assigns to a link $L$ in a solid torus (aka a thickened annulus) a tri-graded abelian group $$\mathit{AKh}(L) = \bigoplus_{i,j,k} \mathit{AKh}^{i,j}(L,k).$$ Grigsby-Licata-Wehrli proved that there's an $\mathfrak{sl}_2(\mathbb{C})$ action on annular Khovanov homology with complex coefficients in which the weight space grading corresponds to the $k$-grading on $\mathit{AKh}.$ It follows that annular Khovanov homology is trapezoidal with respect to the $k$-grading. More precisely, if we let $a_k = \dim\mathit{AKh}(L,k)$, then $$a_k \geq a_{k'}$$ when $k\equiv k'\,\mathrm{mod}\, 2$ and $|k|<|k'|$. In particular,  the sequence $$\{a_k\}_{k\equiv i\, \mathrm{mod}\, 2}$$ is unimodal for any fixed $i$. The question below asks whether something stronger is true:

Question: For a fixed $i$, is the sequence $\{a_k\}_{k\equiv i\, \mathrm{mod}\, 2}$ in fact log-concave?

This is asking whether $$a_{k}^2 \geq a_{k-2}a_{k+2}$$ for all $k$. If so, then this would recover the unimodality above. 

This question is motivated in part by Fox's trapezoidal conjecture, which asserts that the (absolute values of the) coefficients of the Alexander polynomial of an alternating knot form a unimodal sequence; see here for recent progress. Experimental evidence suggests that these coefficients also form a log-concave sequence. The work of June Huh on the log-concavity (hence unimodality) of the coefficients of the chromatic polynomial of a graph is another source of inspiration; see here for a great survey of this and related work.

There is not much experimental evidence for an affirmative answer to the question, merely because no one has computed annular Khovanov homology for very many links (though this is perfectly doable). It would be interesting to gather evidence in favor (or show that the answer is no!) as a first step.

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