  Using the QCD sum rule approach, we show that the flavor-nonsinglet $H$
dibaryon states with J$^{\pi} = 1^+$, J$^{\pi} = 0^+$, I=1 (27plet) are nearly
degenerate with the J$^{\pi} = 0^+$, I=0 singlet $H_0$ dibaryon, which has been
predicted to be stable against strong decay, but has not been observed. Our
calculation, which does not require an instanton correction, suggests that the
$H_0$ is slightly heavier than these flavor-nonsinglet $H$s over a wide range
of the parameter space. If the singlet $H_0$ mass lies above the $\Lambda
\Lambda$ threshold (2231~MeV), then the strong interaction breakup to $\Lambda
\Lambda$ would produce a very broad resonance in the $\Lambda \Lambda$
invariant mass spectrum which would be very difficult to observe. On the other
hand, if these flavor-nonsinglet J=0 and 1 $H$ dibaryons are also above the
$\Lambda \Lambda$ threshold, but below the $\Xi^0n$ breakup threshold (2254
MeV), then because the direct, strong interaction decay to the $\Lambda
\Lambda$ channel is forbidden, these flavor-nonsinglet states might be more
amenable to experimental observation. The present results allow a possible
reconciliation between the reported observation of $\Lambda \Lambda$
hypernuclei, which argue against a stable $H_0$, and the possible existence of
$H$ dibaryons in general.
