Extremal Khovanov homology signature conjecture for Turaev genus one links

Let $L$ be a non-split link of Turaev genus one. Write $Kh^{i,j}(L)$ for its Khovanov homology, let $j_{\min}(L)$ and $j_{\max}(L)$ denote the minimum and maximum quantum gradings supporting Khovanov homology, and let $\sigma(L)$ be its signature. Signature conjecture. At least one of the following statements holds:

1. There is an $i_0\in\mathbb{Z}$ such that

$$Kh^{*,j_{\min}(L)}(L)=Kh^{i_0(L),j_{\min}(L)}(L)\cong\mathbb{Z}$$

and

$$2i_0(L)-j_{\min}(L)=\sigma(L)+1.$$

2. There is an $i_0\in\mathbb{Z}$ such that

$$Kh^{*,j_{\max}(L)}(L)=Kh^{i_0(L),j_{\max}(L)}(L)\cong\mathbb{Z}$$

and

$$2i_0(L)-j_{\max}(L)=\sigma(L)-1.$$

This weakens the hypotheses of the preceding signature theorem, which assumes that the link has both an $A$-Turaev genus one and a $B$-Turaev genus one diagram. The conjecture is open for links that are $A$-Turaev genus one but not $B$-Turaev genus one, and vice versa.