Franz Josif was not really in love with his elder son, and expecially he disliked his wife very much.
Not many know that, in order to marry her, Franz Ferdinand was forced to take an oath regarding denying succession rights to their children.
In practice, Franz Ferdinand was the official heir to Franz Josef, but Franz Ferdinand's sons would not be included in the dynastic succession.
This certainly diminuished very much his importance for Austria-Hungary.
Also, Franz Josif was cernaily not keen in waging a war, even a limited one to Serbia ("I know what a bloodshed war is, I have been at Solferino", he used to say to his prime minister when the latter urged him to take action).
On the other hand (politics is politics) he certainly wanted to gain something from the thing.
Thus, let suppose he explicitly imposes a policy aimed at gaining something from the Sarajevo murder without triggering a war.
What's the maximum that Austria-Hungary could ask?
How to re-formulate the Ulimatum to get it?
What could be the sensible diplomatic step to take?