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.
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