State law does not favor your situation in Louisiana. Your state's law operates under the assumption that the surviving spouse should not inherit anything from the other spouse where there are surviving children.
The other concern is whether the home would be considered community property (hers AND yours) or her separate property. If the property is still in her name from before your marriage to her, then despite your investment of time and money in the improvements, it will likely be considered separate property and pass to her descendants with or without a will devising it to them. In fact, without a will, her nieces and nephews would have a higher priority than you to inherit her property.
If it is held to be community property, which is unlikely, and she dies without a will, you will receive what is called a usufruct which is similar to a life estate in the home and you can continue to live there and use the benefits of it, but when you pass away it will still go to her sons, not your son.
The only way to fix this problem is if she were to execute a deed from herself to you or both of you together, and/or list you in her will as the beneficiary of the home. Otherwise her sons will take the entire property.
Best of luck.
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