Despite the cheating allegations surrounding boyfriend Tristan Thompson, Khloé Kardashian never thought about giving daughter True her own last name

Despite the cheating allegations surrounding boyfriend Tristan Thompson, Khloé Kardashian never thought about giving daughter True her own last name.

“Khloé still very much loves Tristan, and this baby is one half his. True is their child together. Giving the baby his last name was never in question,” a source tells PEOPLE.

Four days after giving birth to her first child, the reality star revealed the couple’s little girl would be taking her father’s last name, despite rumors Thompson, 27, allegedly cheated on the 33-year-old Keeping Up With the Kardashians star throughout her pregnancy.

Want all the latest pregnancy and birth announcements, plus celebrity mom blogs? Click here to get those and more in the PEOPLE Babies newsletter

Screen Shot 2018-04-17 at 8.48.55 AM
Khloé Kardashian’s and Tristan Thompson
| Credit: Khloe Kardashian/Instagram

Tristan Thompson and Khloé Kardashian

Sharing the happy news on Monday, the new mom called her baby girl “such a blessing.”

“Our little girl, True Thompson, has completely stolen our hearts and we are overwhelmed with LOVE,” she wrote on social media, alongside a room filled with pink balloons and flowers. “Such a blessing to welcome this angel into the family! Mommy and Daddy loooooove you True!”

Kardashian also wrote on her website that her daughter arrived weighing 6 lbs., 13 oz., measuring 21 inches in length.

Credit: Khloe With a K

Explaining the origin of the baby’s first name, the reality star’s mother Kris Jenner revealed that the name True is actually a family name.

“I’m so excited to welcome my precious little granddaughter True!!! FUN FACT… my Grandfather’s name on my Dad’s side was True Otis Houghton….my real Dad’s name was Robert True Houghton…so i am so excited Khloe named her daughter True!!! #lovebug #familytradition #family,” the 62-year-old wrote on social media.

RELATED VIDEO: Khloé Kardashian Has ‘Basically Already Forgiven’ Tristan Thompson for Alleged Infidelity: Source

Despite all the infidelity rumors, the source tells PEOPLE that “Khloé really wants to be able to forgive and trust Tristan again and have the life she hoped to have with him.”

“She doesn’t know if she’ll be able to do that, but that’s the goal. She loves the life they had, she loves the baby they have together, and she still loves the dream of being a family together. If she can find a way to rebuild with Tristan, she will,” the source continues, adding that it “still remains to be seen” whether that’s possible.

Credit: Jerritt Clark/Getty Images

Tristan Thompson and Khloé Kardashian

Kardashian’s social media post about her baby name reveal also contained a few hidden signs about the reality star’s current frame of mind regarding her relationship with the NBA player.

“It was very intentional that the statement was basically from the both of them, and reinforces that they are Mommy and Daddy, in this together,” the insider tells PEOPLE.

“The word love was capitalized for a reason too: right now that’s all Khloé has room for her in her heart. This moment is all about love for her,” the insider adds.

While the cheating allegations against Thomspon led fans to speculate whether Khloé’s daughter would still have the surname Thompson, there’s no precedent of the KarJenners deciding not to give their children the father’s last name.

All three of Kourtney Kardashian‘s children with ex-boyfriend Scott Disick share his last name, and the same is true of Kim Kardashian West’s children with husband Kanye West.

New mom Kylie Jenner‘s daughter Stormi also took after her father Jacques Webster — known better by his stage name Travis Scott.

And despite all the relationship issues between Rob Kardashian and ex-fiancée Blac Chyna, the pair also gave their daughter Dream the Kardashian surname, making her Kris’ only grandchild to carry on the Kardashian moniker.