A note on existentially t-henselian fields

A field is existentially t-henselian if it is has the same existential theory in the first-order language of rings as a field that admits a nontrivial henselian valuation. This property turns out to be equivalent to $\mathbb{Z}$-largeness, which is a property identified in previous work with Fehm, and which holds for $F$ if and only if $tF[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$ is not Diophantine in $F(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$, without extra constants. In this short note, we further investigate this property in order to count the number of existential theories of henselian valuations on a given field, and to find other characterizations of existential t-henselianity.