  Given a C$^*$-algebra $A$, let $S(A^+)$ denote the set of those positive
elements in the unit sphere of $A$. Let $H_1$, $H_2,$ $H_3$ and $H_4$ be
complex Hilbert spaces, where $H_3$ and $H_4$ are infinite-dimensional and
separable. In this note we prove a variant of Tingley's problem by showing that
every surjective isometry $\Delta : S(B(H_1)^+)\to S(B(H_2)^+)$ or
(respectively, $\Delta : S(K(H_3)^+)\to S(K(H_4)^+)$) admits a unique extension
to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$
(respectively, from $K(H_3)$ onto $B(H_4)$). This provides a positive answer to
a conjecture posed by G. Nagy [\emph{Publ. Math. Debrecen}, 2018].
