The owner of a possesses the same interest as the owner of a fee simple absolute, except that this interest is subject to a condition that may result in termination of the interest.

Conditional Fee Estate

Explanation:

Based on the information provided within the question the term being described in the question is called a Conditional Fee Estate. This is (like mentioned in the question) similar to a fee simple except the property must eventually be passed on to the heirs. Which is why it is named as Conditional, because failure to do so will eventually cause the termination of the interest.

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