Bessa [Be] proved that for given
$n$
and
${{i}_{0}}$
, there exists an
$\varepsilon (\text{n,}\,{{i}_{0}})\,>\,0$
depending on
$n$
,
${{i}_{0}}$
such that if
$M$
admits a metric
$g$
satisfying
$\text{Ri}{{\text{c}}_{(M,g)}}\ge n-1,\text{in}{{\text{j}}_{(M,g)}}\ge {{i}_{0}}\,>\,0$
and
$\text{dia}{{\text{m}}_{(M,g)}}\,\ge \,\pi \,-\,\varepsilon $
, then
$M$
is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.