A bound on the $\mu$-invariants of supersingular elliptic curves

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve and let $p$ be a prime of good supersingular reduction. Attached to $E$ are pairs of Iwasawa invariants ${\mu }_p^{\pm }$ and ${\lambda }_p^{\pm }$ which encode arithmetic properties of $E$ along the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that ${\mu }_p^{\pm }=0$. We provide support for this conjecture by proving that for any $\mathrm{\ell }\ge 0$, we have ${\mu }_p^{\pm }\le 1$ for all but finitely many primes $p$ with ${\lambda }_p^{\pm }=\mathrm{\ell }$. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that ${\mu }_p^{\pm }\le 1$ holds on a density 1 set of good supersingular primes for $E$.