  We show that a compact quaternionic contact manifold of dimension seven that
satisfies a Lichnerowicz-type lower Ricci-type bound and has the $P$-function
of any eigenfunction of the sub-Laplacian non-negative achieves its smallest
possible eigenvalue only if the structure is qc-Einstein. In particular, under
the stated conditions, the lowest eigenvalue is achieved if and only if the
manifold is qc-equivalent to the standard $3$-Sasakian sphere.
