In the post (cross-posted in Mathematics Stack Exchange with identificator MSE 4244256 and same title) we assume that $P(x)=a_0+a_{1}x+\ldots+a_{n-1}x^{n-1}+a_{n}x^n$ is a polynomial of degree $1<\deg(P)=d=n$ defined over a field $K$ of characteristic zero. We denote its derivatives as $P^{(i)}(x)$ (writting $P^{(0)}(x)=P(x)$), and $a_n$ denotes the leading coefficient of $P(x)$.

I've stated two conjectures (as speculations from the fact) inspired in that I've proven inductively that each polynomial $p(x)$ of degree $1<\deg(p)$ (and with corresponding $a_n\neq 0$) satisfies $$p(x)=a_n\cdot\left(\frac{n-l}{\frac{d}{dx}\log p^{(l)}(x)} \right)^n\tag{1}$$ for each integer $l$ with $0\leq l<n-1$.

Conjecture 1. Let $P(x)$ a polynomial of degree $1<n$, thus we assume $P^{(n)}(0)\neq 0$. If the equation $$P(x)=\frac{P^{(n)}(0)}{n!}\cdot\left(\frac{n-l}{\frac{d}{dx}\log P^{(l)}(x)} \right)^n$$ holds for each integer $0\leq l<n-1$, then $P(x)$ has the form $$P(x)=\frac{P^{(n)}(0)}{n!}\cdot(x-\alpha)^n$$ for some element $\alpha\in K$.

I know the statement of the called Casas-Alvero conjecture from Wikipedia Casas-Alvero conjecture. I've speculated (while I don't know if this has a good mathematical content, or if these ideas are in the literature in some way more or less explicit) if from previous simple idea one can to state an equivalent form of Casas-Alvero conjecture.

Conjecture 2. Let $P(x)$ a polynomial of degree $1<n$ and leading coefficient $a_n\neq 0$. The Casas-Alvero conjecture is equivalent to that the equation $$P(x)\cdot\left(P^{(l+1)}(x)\right)^n=a_n\left((n-1)P^{(l)}(x)\right)^n,\tag{2}$$ holds for each integer $0\leq l<n-1$.

Question. I would like to know what work can be done about the veracity of previous conjectures: can you prove or refute (providing a counterexample, or reasoning if my conjectures have not a good mathematical content) these conjectures Conjecture 1 and Conjecture 2? Many thanks.

I don't know if these conjectures Conjecture 1 and Conjecture 2 are in the literature, my idea was very simple: that is to study the logarithmic derivative of derivatives of a given polynomial of the cited form, if it is in the literature please refer it in a comment or in your answer and I try to search and study this from the corresponding articles.

Remark: recently a professor solve a related question [1].

References:

[1] Iterated derivatives and polynomials that are the power of a linear polynomial, Mathematics Stack Exchange (Sep 12, 2021), post with identificator MSE 4248459