The South was still two and a half years away from realizing their cause was lost. Lincoln was under no illusion that the war was close to being over.
I've seen it asserted that the Emancipation was primarily a tool for weakening the South's attempts to get foreign support and to boost the moral of his supporters, plus laying the political ground work for stealing all the money the South invested in slavery by taking them without recompense.
I think there were several reasons for the Emancipation Proclamation. Certainly it was for foreign consumption. It was an economic tool against the South. And it prepped the way for total emancipation.
Beside the point. Had they done so, Lincoln would have likely taken the deal. At least that is what he says when he wrote that letter.
Of course there is always the possibility that he might be a two faced liar, but I wouldn't think you would find that a likely possibility.
If you think Lincoln would have kept his word, the US would have likely kept slavery had the South conditionally agreed to stop fighting as of August 22, 1861.
And it prepped the way for total emancipation.
It got everyone ready for a vast theft of all their invested wealth.