Is my insurance going to increase after putting in a claim that a tow truck company caused to my car?

Question Details: A towing company damaged my car while towing it when there was a street cleaning and I did not get to move my car in time. They acknowledged that it was their fault, but instead of the company dealing with the expenses, the driver who damaged my car is the one who's supposed to pay me. He obviously does not want to pay the $2500 in damage, ignores my questions as to why he's the one paying for damage instead of the company, and instead wants to pay my deductible of $500. However, won't this cause my insurance to go up?

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