Should we expect an approximate version of Euler’s Sum of Powers Conjecture to hold for higher power equations?

Consider the Diophantine equation:

$$x_1^k+x_2^k+\dots+x_n^k=y^k.\tag 1$$

For any exponent $k$, let $N(k)$ be the smallest possible number of terms on the left, $n$, such that there exists a non-trivial solution in positive integers. Euler’s Sum of Powers Conjecture is equivalent to the statement that, for any $k$, $N(k) \geq k$. As is well known, the Conjecture is false, since $N(4) = 3$ (source: Mathworld – Diophantine Equations 4th powers, equation (7)).

Nevertheless, it seems plausible to suppose that there is some looser relation between $k$ and $N(k)$. Two reasons can be given. One is the pattern of known smallest $n$ for given $k$ (source Mathworld – Diophantine Equation $n$th Powers):

In each of these cases, except $k=10$, we have $k-1 \leq n \leq k+1$. However, the force of this observation is diminshed by the fact that these numbers are based only on currently known solutions. Only for $k \leq 4$ is it certain (from the Fermat-Wiles Theorem) that no smaller $n$ is possible. So far as I am aware, it has not been proved impossible that $N(5) = 3$, or that $3 \leq N(6) \leq 6$, or similarly for higher $k$.

A second reason for the plausibility of a looser relation is that increases in $n$ and increases in $k$ might be expected to have opposing effects on $N(k)$.

Effect 1 Each addition to $n$ creates one extra degree of freedom which increases the number of possible sums of the left hand terms and so (other things being equal) tends to reduce $N(k)$ for given $k$.

Effect 2 Each addition to $k$ Increases the spacing between successive $k$th powers and so (again, other things being equal) tends to increase $N(k)$ for given $n$.

Question

For exponents $k > 10$, which of the following scenarios can be expected to emerge:

A) Effects 1 and 2 above broadly offset each other, so that we can expect $N(k) \leq k$ for very approximately half of all exponents?

B) Effect 1 tends to dominate, so that we can expect that $N(k) \leq k$ for most $k$?

C) Effect 2 tends to dominate, so that we can expect that $N(k) > k$ for most $k$?

Note that the question asks only about the existence of solutions. It is not concerned with whether solutions are currently known or within the range of current computer searching. Also, it is limited to equations with a single right hand term, so it is distinct from the Lander, Parkin & Selfridge Conjecture, which allows multiple terms on both sides.

Note: Why “other things being equal”?

If the possible values of the left and right hand sides of (1) were randomly distributed within the relevant ranges, then it might be expected that Effects 1 and 2 alone would determine the likelihood of a solution for given $n$ and $k$. But clearly they are not random. One reason is that for any $k$, the $y^k$ are more widely spaced for larger than for smaller $y$. Another is that, if $k=p-1$ for some prime $p$, then Fermat’s Little Theorem implies that for any $x$, $x^k \equiv 0,1 \pmod{k+1}$, which imposes a significant constraint on possible values. There are other modular constraints. We should therefore expect Effects 1 and 2 to result only in a broad trend in the relation between $N(k)$ and $k$, with departures from the trend at some particular values of $k$.