Quick answer: zero — Alaska charges no state capital gains tax at all, on this sale or any other. See the full cost picture in the Alaska probate real estate sale calculator.
No state income tax, full stop
A rare election that can shrink the federal gain too
A married couple who elected community property in Alaska gets a full step-up in basis on the whole property at the first spouse's death — not just the deceased spouse's half, the way most common-law states work — which can shrink or eliminate the federal gain on an eventual sale.
A federal step-up still matters either way
The house typically still receives a federal stepped-up basis to its value on the date of death, often leaving little or no federal capital gain if the house sells reasonably soon after — frequently the bigger factor than any state-level rule.
No estate or inheritance tax layered on top
Alaska imposes neither, so only the federal estate tax, with its much higher threshold, can apply to larger estates.
A local probate attorney can review your estate — many offer a free consultation.
The zero-income-tax rule applies identically no matter which Alaska borough or census area the sale closes in — Alaska has no counties at all, unlike most other states.