Surprising GOP gains in House races Tuesday and the addition of women to their caucus appears to have cleared the field of challengers to House Minority Leader Kevin McCarthy, House leadership sources tell Axios.Why it matters: That clears the path for him to run for speaker of the House in 2022 if Republicans are successful in a push to retake the majority in the midterms. Driving the news: Three Republican leadership sources said neither House GOP Conference chair Liz Cheney nor Republican whip Steve Scalise will mount a bid against McCarthy, and that all three will seek to hold their current minority roles.Between the lines: Tuesday's election was a far more successful night for House Republicans than expected; earlier predictions estimated they could lose 10-15 seats. That has emboldened the conference and solidified McCarthy's standing.Ahead of the election, many Republicans believed a loss of seats could prompt the party to reexamine its leadership.A spokesperson for McCarthy declined to comment.The latest: In a letter to the House GOP conference on Wednesday, McCarthy wrote, "Although securing an outright majority remains our aim, we can take many positives from last night and use the places we came up short as fuel to motivate our efforts over the next two years.""I firmly believe this conference has what it takes to earn back the majority. More than that, I am a fighter and I do not quit on my team or on our shared goals. That is why I am running to serve as Republican Leader and humbly ask for your support."This story has been updated to include McCarthy's letter to the House Republican conference.