Definable v -topologies, Henselianity and NIP

We initiate the study of definable V-topologies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if (K,v₁,v₂) is a bi-valued NIP field with v₁ henselian (respectively, t-henselian), then v1 and v₂ are comparable (respectively, dependent). As a consequence, Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.