Try Neem Oil since it is organic and easy to find. The one recommended by a commercial grower is Milstop. You can find it online but you may balk at the price but a five pound bucket goes a long way.

I wouldn't think you would be having fungus problems this time of year unless you have it constantly wet and in deep shade. The other thing you may want to do is remove all bad looking leaves which will encourage the plant to put out new leaves.

Again to my rescue! The plant lives with other begonias indoors in a bright, north-facing window but I mist them a couple of times a day, but only this cane has been affected. I'll pull off the worst leaves and Neem the others.

Thanks again!