  We have systematically calculated the strong decays of the low-lying
$\rho$-mode $1P$-wave $\Lambda_{c(b)}$, $\Sigma_{c(b)}$, $\Xi_{c(b)}$,
$\Xi^{'}_{c(b)}$, $\Omega_{c(b)}$ baryons using the chiral quark model within
the $j$-$j$ coupling scheme. For the controversial states, our results
indicate: (i) For the singly charmed heavy baryons, the newly observed
$\Lambda_{c}(2910)^+$ is a good candidate of the $J^P=5/2^-$ state
$\Lambda_c|J^{P}=\frac{5}{2}^{-},2\rangle_{\rho}$. (ii) For the singly bottom
heavy baryons, the $\Xi_{b}(6227)^-$ favors the $J^{P}=5/2^{-}$ state
$\Xi_{b}|J^{P}=\frac{5}{2}^{-},2\rangle_{\rho}$. (iii) The other missing
$\rho$-mode $1P$-wave excitations in $\Lambda_{b}$, $\Sigma_{c(b)}$ and
$\Xi^{'}_{c(b)}$ families appear to be broad structures with
$\Gamma$$\sim$(100-200) MeV, and their strong decay widths are sensitive to
their masses. (iv) The $\rho$-mode $1P$-wave $\Xi_{c(b)}$ and $\Omega_{c(b)}$
baryons have a relatively narrow decay width of a few MeV or a few tens of MeV,
and have a good potential to be observed in forthcoming experiments.
