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Converse implication from free functional Santaló to SSFTI

Establish that the free functional Blaschke–Santaló inequality implies the sharp symmetrized free Talagrand transport–entropy inequality (SSFTI) in its maximal correlation formulation.

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Background

The paper proves one direction (that SSFTI implies the free functional Santaló inequality) and discusses the duality known in the classical case. However, the converse implication in the free setting remains unsettled.

Clarifying this converse would complete the duality between transport–entropy inequalities and functional Santaló-type inequalities in free probability, mirroring the classical theory.

References

However, it is still unclear whether the converse holds in full generality: Does the free Blaschke-Santal´o functional inequality imply (SSFTI) in its maximal correlation formulation?

— A sharp symmetrized free transport-entropy inequality for the semicircular law (2410.02715 - Diez, 3 Oct 2024) in Remark 13, Section 3