Strict inner-function inequality conjecture in Hardy spaces

Let $H^p$ denote the Hardy space on the unit disk, let $d54b$ be the unit circle, and let $f\in H^p$ satisfy $f(0)\neq 0$. An inner function is a bounded analytic function on the unit disk whose radial boundary values have modulus one almost everywhere. Strict inner-function inequality conjecture. If $J$ is any non-constant inner function, then there exists a constant $c=c_f\in\mathbb{T}$ such that

$$\|1-Jf\|_p>\|1-cf\|_p.$$

Such an inequality would strengthen the preceding proposition and would imply that non-trivial optimal polynomial approximants in $H^p$ cannot vanish in the open unit disk. The supplied text gives no resolution of the conjecture.