Questions related to the Mahler and the Viterbo Conjecture

A question that I'm very much interested in is: for \( K\subset \mathbb{R}^n \) a centrally symmetric convex body with polar dual body \( K^{\circ}\subset \mathbb{R}^n \) and \(c\) any normalized symplectic capacity, is it true that

 \(\frac{c(K\times K^{\circ})^n}{n!}\leq Vol(K\times K^{\circ})\)

holds? This is part of the Viterbo isoperimetric conjecture and it is known (by a result of Arstein-Avidan, Karasev and Ostrover) to imply the Mahler volume conjecture. While the original and more general statement of the Viterbo conjecture has now been proven wrong (by Haim-Kislev and Ostrover), there's still hope that the less general case stated above does hold. Here's a list of related questions in this topic:

1. Is it true that \(c_{Gr}(K\times K^{\circ})=4\) for any \(K\) and \(c_{Gr}\) the Gromov width? This immediately implies the Mahler conjecture, since the Viterbo conjecture is easily seen to hold for the Gromov width. This has been shown to be true for \(K\) being and ellipse or a cube.
   
   
2. For \(n=2\), is \(K\times K^{\circ}\) symplectomorphic to a toric domain? If so, is it a monotone, or even better, concave toric domain? (A positive answer to this would give a new proof of the Mahler conjecture in dimension 2 by a result of Gutt, Hutchings and Ramos) This has been shown for \(K\) being a ball (by Ramos) and a square (by Ramos and Sepe). A first, more modest attempt, would be to assess this for \(L^p\)-balls. I do not think this is true in higher dimensions, for once, in work in progress with Vianna and Achig-Andrango, we construct singular Lagrangian fibrations for the Lagrangian bidisk in dimension 3 and show, using Hutchings zeta function, that it cannot be symplectomorphic to a toric domain.
   
   
3. In my paper The strong Viterbo conjecture and various flavours of duality in Lagrangian products two more kinds of dual Lagrangian products are proposed, that is, let \(\Phi\) be a \(n\)-tuple of Young functions with Legendre transform \(\Phi^*\) and \(K_{\Phi}\) the unit ball for the Luxemburg metric induced by \(\Phi\). We can consider the usual polar dual Lagrangian product \(K_{\Phi}\times K_{\Phi}^{\circ} \), but we can also consider the ``dual functional" Lagrangian product \(K_{\Phi}\times K_{\Phi^*} \) and ``dual polar-functional" Lagrangian product \(K_{\Phi}\times K_{\Phi^*}^{\circ} \). For the first of these two new classes, it was shown that all normalized symplectic capacities agree and for the usual dual Lagrangian product, some cases where all normalized symplectic capacities agree were identified. Nothing is known for the second new class. It would be interesting to know whether all normalized symplectic capacities agree there too.
   
   
4. Are the new classes in Question 3 maybe symplectomorphic to toric domains for \(n=2\)? I have no reasons to expect this to be true at the moment, but I do think is an interesting question and possibly easier to tackle than Question 2, since there may be a chance that an integrable system can be constructed out of \(\Phi\) and \(\Phi^*\). 
   
   
5. A weakening of Question 2 is whether billiard orbits (except gliding orbits on the boundary) on the Minkowski table \(K\) with geometry given by \(K^{\circ}\) come in \(S^1\)-families? This is easily seen to be true for 2-bouncing orbits. Notice this is a necessary condition for a (possible) positive answer of Question 2.