Your opponents last hand starts as follows:
Platinum, Cartographer, Cartographer, Remodel, Copper
The Cartographers draw a Gold and Taxman leaving a Baker and Gold as the top 2 cards of the deck.
Now comes the decision. Your opponent has Platinum, Gold, Remodel, Taxman, Copper in hand. At this stage in the game, you need green and your opponent has a couple of options.
First - Remodel Platinum --> Colony and buy Estate netting 11 points but leaving 1 Colony left
Second - Remodel Taxman --> Duchy and buy Province netting 9 points but keeping the Colony stack at 2.
Your last hand is Cartographer, Cartographer, Copper, Forge, Colony. Regardless of what option your opponent takes you can not win on that turn. Your Cartographers will draw at least 1 good card in your hand, but there is no way to get Colony + Duchy or Double Colony (buy one/forge one) depending on opponents decision. Your opponent is on the cusp of reshuffling and has Gold/Baker as the next two cards. Best case scenario for you on your turn is draw Province/Duchy or Colony/Naught. You will likely be able to set up your next hand, but your opponent is in the driving seat in terms of penultimate victory cards and does have a deck that can make it.
I'm going to have to give this one to your opponent in a nail biter. Especially if they choose option 1. With your deck, you are unlikely to get Colony/Duchy to make up the 1-point difference.