The two-fixed-points-or-high-periodic-points conjecture for torus homeomorphisms

Let $F\colon \mathbb{T}^2\to\mathbb{T}^2$ be a homeomorphism homotopic to the identity and satisfying the curve intersection property. Under these hypotheses, either $F$ has two fixed points or $F$ has periodic points of arbitrarily high periods.

Two-fixed-points-or-high-periodic-points conjecture. Either $F$ has two fixed points or $F$ has periodic points of arbitrarily high periods.

This is presented as the expected strengthening of the preceding theorem: when the curve intersection property holds, a torus homeomorphism either has rich periodic dynamics or, in the fixed-point case, should have a second fixed point. The supplied text does not establish the assertion, and its resolution is not specified.