Is the following theorem known, or can be easily derived from known results?

Consider the differential equation $$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$ where $k>0$ is fixed, $\lambda$ is a large (complex) parameter and $\phi$ is a complex valued function analytic on $[0,1]$. It is known that there exists a unique solution $w_0(z,\lambda)$ which is analytic at $z=0$ and $w_0(0,\lambda)=1$. It is also known that $f(\lambda)=w_0(1,\lambda)$ is an entire function.

Theorem (?). $f(\lambda)=(1+o(1))\exp\sqrt{\lambda},$ as $\lambda\to\infty$, $|\arg\lambda|\leq\pi-\epsilon$, where $\sqrt{\lambda}$ is the principal branch.

Remark. There is another solution, normalized by $w_1(z,\lambda)\sim z^{k+1},z\to 0$, and for this solution I know how to prove the result, and know the references, for example Olver, Asymptotics and special functions, Ch. 12.

More remarks: 1. When $\phi=0$ this reduces to Bessel's equation; the Theorem is true in this case but not trivial.

In the definition of $w_0$ the crucial word is ANALYTIC. There are infinitely many other solutions $w$ satisfying $w(0)=1$: adding to $w_0$ any multiple of $w_1$ does not change the value at $0$. And the conclusion of the Theorem does NOT hold for some of solutions satisfying $w(0)=1$. For example, when $\phi=0$ there is a solution with $w(0)=1$, for which $w(1,\lambda)$ decays exponentially for $\lambda>0$.

The most important case for me is when $\phi$ is even, which may help. But on my opinion, the problem is interesting in the general case as well.

Remark. Now I proved this, but the question remains whether this follows from some known, published results.