Non-torsion degree-zero line bundles on abelian varieties are pt-ample (but not tensor-ample)

Prove that for any abelian variety A and any line bundle L of degree 0 with L^n ≄ O_A for all n>0, the induced polarization τ_L on Perf(A) is pt-ample while L is not ⊗-ample.

Background

Motivated by the elliptic curve case, where non-torsion degree-zero line bundles yield pt-spectra equal to the tt-spectrum but are not ⊗-ample, the authors propose an abelian-variety-wide generalization.

This conjecture clarifies the difference between pt-ampleness and ⊗-ampleness in the context of polarizations and spectra.