Large-load uniqueness conjecture for the canonical dual problem

Let $({\cal P})$ be a properly posed problem satisfying Assumption 1, let $\mathbf f$ be its load vector, and let $({\cal P}^d)$ be the associated canonical dual problem over ${\cal S}^+_c$. Large-load uniqueness conjecture. There exists a constant $f_c>0$ such that $({\cal P}^d)$ has a unique solution in ${\cal S}^+_c$ whenever

$$\|\mathbf f\|\geq f_c.$$

The claim asserts that sufficiently large loads force uniqueness of the canonical dual solution. The supplied text gives no evidence of a proof or disproof, and the precise content of Assumption 1 is not included, so the conjecture remains open.