Spherical braids with the same knot Floer homology

Question: are there infinitely many spherical braids in $S^1\times S^2$ with the same knot Floer homology?

There is a simple way of constructing infinitely many knots in $S^3$ with the same knot Floer homology: take a nontrivial band sum of two unknots; adding twists to the band preserves knot Floer homology (see Theorem 1 of Hedden--Watson). The family of genus-2 fibered hyperbolic pretzel knots $P(-3,3,2n)$ can be described in this way. 

The existence of this family of pretzels shows that the knot Floer homology of a hyperbolic fibered knot in $S^3$ cannot in general bound the dilatation of its pseudo-Anosov monodromy. Indeed, a hyperbolic fibered knot is determined by the conjugacy class of the pA representative of its monodromy, and there are only finitely many conjugacy classes of pA homeomorphisms of a given surface with dilatation less than some fixed constant. (One can also conclude that infinitely many of the pretzels in this family must have monodromies which are not strongly irreducible, by Aougab--Futer--Taylor.)

The question above is asking whether a similar phenomenon occurs for spherical braid knots in $S^1\times S^2$. These are the analogues in $S^1\times S^2$ of fibered knots in $S^3$. This class of knots is interesting as all knots in $S^1 \times S^2$ with longitudinal L-space surgeries are known to be spherical braids, by Ni--Vafaee. 

A negative answer would indicate that the knot Floer homology of a spherical braid knot might bound the dilatation of the braid monodromy. If so, this would show that there are only finitely many knots in $S^1 \times S^2$ with longitudinal surgeries in each homology class, an analogue of the conjecture that there are only finitely many L-space knots in $S^3$ in each genus. A positive answer, which seems more likely, would also be interesting.

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