33k7
The other factor is that until the conflict Bulgaria was Russia's main ally in the Balkans. It was anger at Russia's perceived lack of support that meant relations between the two worsened and Russia found Serbia was its only real ally in the region.
If this doesn't happen Russia doesn't feel anything like as compelled to support the Serbs, especially since the conservative elite will be deeply alienated by the assassination of a royal and the dangerous precedence set. Hence Serbia might buckle under the Austrian threat or, if Austria does attack it might not prompt a general war.
In that case Austria gets a nasty shock as it would probably see a serious defeat in the first set of offensives as OTL. Very likely to wear Serbia down eventually, or possibly, if Serbia has no support, some compromise peace is agreed.
Alternatively, Bulgaria, since it still wants the Macedonian lands promised [it believed] in the initial alliance, might join the war, possibly even with Russian support. In which case Serbia probably goes down quickly and Bulgaria gets land but also is rather exposed. [Since it has land Serbia, Romania and Turkey want and also outstanding differences with Greece]. There is the chance, depending on Russia reaction to Bulgaria joining such a conflict, it might fall out with them and end up at least loosely allies with the central powers.
This of course assumes that the F-F assassination isn't butterflied, which is all too likely.

However if it isn't and TTL the 2nd Balkan War is Austria [possibly with Bulgaria] defeating Serbia then you might not get a great war. I say this because I think a window for it, especially with the current alliances, is reality short. Between the declining position in the naval race, growing concern about the S plan and also the steadily growing strength of Russia Germany has only a short period where it can even hope to win a general war. With Austria tied down for several months to a year and showing serious shortcomings it won't be available to assist and won't look that strong an ally while the Austrian leadership may also want some time to re-organise their military at least.
Steve