It almost definitely would - the Mongols had set their sights on Persia and the Abbasid Caliphate, and they had already organized several earlier campagins in this area.
You misunderstand me.
I am not asking if Mongols would have reason to invade, I am asking if they would have men, money and leaders.
We have jsut agreed thet 130 000 Mongols went off to Europe. They won't all be coming back. And whatever you might say about using auxiliaries, they won!t have Europeans in 1450's in their armies yet.
Also conquest of Europe woudl divert lot of capital. Not only for supporting the troops, but for setting up new administration. I find it somewhat doubtfull that European conquest is going to be profitable right away in large scale. Certainly Mongols will get plunder and tributes from fearful lords, but that would likely remain in hands of Mongols in the place, enriching the conquerors.
And if the Mongol conquerors of Europe get good money for themselves, if not for whole Khanate. Than we can expect a flow of ambitious Mongols into Europe in addition to any other distractions they akready had in this period.
Further- If Outremer is strengthened by refugees from Europe (who in the short time will be more of a burden), it will cause a reaction from Muslims who will have to come together in face of this threat.