Can I replace "to conquer" by "conquering" in this sentence?

The gerund and the infinite equate to the continuous and simple forms of the verb.

All simple forms of the verb indicate an action as a whole - from start to finish.

The simple form of the verb can indicate a habitual or regular action that

(i) is/was/will be complete/completed each time it is undertaken.

A: What do you do to keep fit?

B: I ride a bike. -> “ride” includes everything from getting on the bike at the start of the journey to getting off the bike at the end.

Or

(ii) a single, complete or completed present, future, or past action:

"He told me that I had to visit the Eiffel Tower, so I go/went/will go to Paris on Wednesday” -> “go/went/will go” includes everything from the decision being made, bags being packed, going to the airport, etc., to the arrival in Paris.

(iii) a habitual, recurring, regular or frequent action (that is completed each time)

On Saturdays, I go to the gym.

He ate toast for breakfast every day of his life.

The continuous form of the verb indicates

(i) an action that is/was/will be (a) incomplete and (b) in progress and (c) at the time that is being referred to (it has started but it has not yet finished) ->

I will be/am/was/have been/had been riding a bike = I will be/am/was/have been/had been in the process of riding a bike but have not yet finished riding the bike at the time I am referring to.

The continuous form used to be known as “the imperfect”: It was called “imperfect” because the action had not been “perfected” i.e. it had not finished.

OED

Imperfect: 5. Grammar. Applied to a tense which denotes action going on but not completed; usually [edit Q- but not always] to the past tense of incomplete or progressive action.

1871 H. J. Roby Gram. Latin Lang. §549 Three [tenses] denoting incomplete action; the Present, Future, and Imperfect (sometimes called respectively, present imperfect, future imperfect, past imperfect).

According to this source:

Gerunds are often used when actions are real, fixed, or completed. "I enjoy cooking." (emphasis added)

Infinitives are often used when actions are unreal, abstract, or future: "He wants to swim." (emphasis added)

I would say the notion of conquering is an abstract action, so the infinitive would be the correct choice.