That reason would be awful, and most likely the consequences too, but it's not readily obvious to me that the timing would be.
The North marching South would get little to no support from their allies and would be repelled rapidly before returning to the current status quo.
So yes, I am sure. Neither sides allies have any desire to be the agressor. Not to mention that simply starting a war to cling on to power might be the stupidest reason there is to start a war.
What if they didn't do that (at first), but instead (as an intermediate step) annexed the north as a colony with its own economy but South Korean governorship; and then opened said colony to worldwide foreign aid?
This would be very similar to the short-term arrangement negotiated between West Germany and East Germany after the fall of the Berlin wall. For analogous reasons, West Germany didn't want to immediately absorb/unify with East Germany; instead, they held political and military stewardship over East Germany until it could be built back up.
It's one of the richest country in the world, right next to one of the poorest. Unification in any sense at this point is suicide by the South. The misery inflicted upon the southern Koreans will be immeasurable if both countries are unified. The leaders of both Koreas know this.
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Or, alternately, given that literally everything in NK would need to be rebuilt at some point... rather than "swallowing the elephant whole", they could take incremental bites: annex a few miles of North Korea into themselves; revitalize/gentrify that part; repeat. Slowly, over 50 years.
(How? At the end of the war, they'd do a census, figuring out where every NK citizen lived as of the end of the war. Then, for each annexation, anyone already living in the annexed area as of the end of the war, would automatically become a South Korean citizen. Anyone who moved to the area after the end of the war would not, and would instead be required to vacate northward, with the SK government compensating them for the [very low value of the] home they're being displaced from, per eminent domain.)
Of course, any NK citizen would be free to apply at any time for a South Korean visa — just like a citizen of any other country can. They wouldn't be especially prioritized.
And an NK citizen would also be equally free to apply for a visa to move to literally anywhere else — possibly as a refugee. SK would probably encourage this and even assist with it.
By a curious geographical coincidence, neither North Korea nor North China nor North Post-USSR are known for their independent reality-based reporting and I take it as a given that North Korea would never start a war of aggression. If the North should ever march South (again) it will of course be as a completely righteous response against intolerable provocation and aggression from its eternal enemy, as confirmed by media duly approved by Moscow and Beijing.
They're looking more ready and eager to do that by the month, and with the present Global political situation, the respect for current status quo seems to be at a low.
If you perceive the respect (NK has) for current status quo is at a low then it is most likely that NK desperately needs food or fuel for the coming winter, for every other player in the world/region the desire for the status quo is at an all time high.
