Yes, I think it is much more an issue if the sister initiates the divorce, because the organization places a very high importance on keeping marriages together. It can work both ways, though.
A friend of mine was married to a sister who cheated on him and had a daughter with another man. She went to the elders and confessed, and my friend agreed to forgive her and even raise the daughter as his own (her biological dad showed -and continues to show- little interest in her). A couple of years later, his wife again went to the elders to confess... to several more affairs.
Officially, the elders took no action and advised him to forgive her. Privately, one of them told my friend that he stood behind the advice they gave him, to try to salvage the marriage. Yet, he admitted that he would likely not do so himself, under the circumstances. My friend filed for divorce and I don't think he was ostracized at all. Then again, he is a nice person and very helpful to everyone, and the circumstances (repeated infidelity with multiple partners and no corrective action taken) make him a sympathetic figure.
The desire to avoid bringing reproach on Jehovah's name can cause elders to be rigid and unbending, even when they are confronted with a unique situation. Which can end up bringing the reproach that they were hoping to avoid. His ex-wife is out for many years now, though she hung on for some time. It's not a good look for the congregation.