I have not yet discussed these concerns about the consistency of I3-I0 cardinals with many set theorists. Please upvote if you think my concerns about the algebra of elementary embeddings should produce at least some doubts about the consistency of rank-into-rank cardinals. Please downvote if you do not think my concerns about the algebras of elementary embeddings should produce doubts about the consistency of rank-into-rank cardinals. I have made this post community wiki so that people will feel more free to upvote or downvote this answer based on these new criteria.
I personally have some doubts about the consistency of rank-into-rank cardinals since I am concerned about a possible inconsistency arising from the algebras of rank-into-rank embeddings.
Define the Fibonacci terms $t_{n}$ for $n\in\omega$ by $t_{1}(x,y)=y,t_{2}(x,y)=x$ and $t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$. A reduced permutative LD-system [1] is a left-distributive algebra $(X,*)$ together with an element $1\in X$ such that $1*x=x$ for each $x\in X$ and where for each $x,y\in X$ there exists an $n$ with $t_{n}(x,y)=1$.
Intuitively, the reduced permutative LD-systems resemble the algebras of elementary embeddings $V_{\lambda}/\equiv^{\gamma}$ in the following ways:
The reduced permutative LD-systems have a notion of a composition operation. In particular, if $x,y\in X$, then we can define $x\circ y=t_{n+1}(x,y)$ where $n$ is a natural number such that $t_{n}(x,y)=1$.
The reduced permutative LD-systems have a notion of a critical point. We define $\textrm{crit}(x)\leq \textrm{crit}(y)$ if there exists some $n$ where $x^{n}*y=1$. The notion of a critical point is very well behaved for reduced permutative algebras since critical points in reduced permutative LD-systems satisfy most of the main properties that critical points of rank-into-rank embeddings satisfy. Furthermore, the notion of a critical point in reduced permutative LD-systems is not just a generalization from set theory to algebra but the notion of a critical point in a reduced permutative LD-system is an essential part of the theory of permutative LD-systems.
Every algebra of the form $V_{\lambda}/\equiv^{\gamma}$ is a reduced permutative LD-system.
At this point, it is reasonable to conjecture that every finite reduced permutative LD-system is isomorphic to some subalgebra of some $\mathcal{E}_{\lambda}/\equiv^{\gamma}$.
$\mathbf{Fact:}$If $j:V_{\lambda}\rightarrow V_{\lambda}$ is elementary, then $j*j(\alpha)\leq j(\alpha)$ for all $\alpha<\lambda$. In particular, $\textrm{crit}((j*j)*k)=(j*j)(\textrm{crit}(k))\leq j(\textrm{crit}(k))=\textrm{crit}(j*k)$ for all $j,k:V_{\lambda}\rightarrow V_{\lambda}.$
On the other hand, there exists permutative LD-systems $(M,*)$ along with $x,y\in M$ such that $\textrm{crit}((x*x)*y)>\textrm{crit}(x*y)$ (such algebras $(M,*)$ were discovered using computer calculations). Therefore, the algebra $(M,*)$ cannot arise from the algebras of elementary embeddings. One possible explanation between this discrepancy is that may be possible to show that $(M,*)$ is actually a subalgebra of some $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ in some model and thus obtain an inconsistency.
The algebras $M$ where $\textrm{crit}((x*x)*y)>\textrm{crit}(x*y)$ for some $x,y\in M$ together with the great consistency strength of rank-into-rank cardinals has given me some doubts about the existence and consistency of rank-into-rank cardinals. After all, the rank-into-rank cardinals are very close to the Kunen inconsistency, and they are far above the cardinals for which there exists a good inner model theory. Furthermore, the mere fact that rank-into-rank cardinals may be used to prove purely algebraic results is a reason to believe that rank-into-rank cardinals are the most vulnerable spot of the large cardinal hierarchy to an inconsistency.
With everything being said, there is likely a better explanation for the existence of reduced permutative LD-systems $(M,*)$ with $\textrm{crit}((x*x)*y)>\textrm{crit}(x*y)$. It is likely that the only reason there seems to be a discrepancy between algebra and set theory is that the algebras of rank-into-rank embeddings are very poorly understood. When the algebras of rank-into-rank embeddings become better understood, I will likely recant my doubts about the existence and consistency of rank-into-rank cardinals. Lastly, the near inconsistency of rank-into-rank cardinals seems to imply that algebras of rank-into-rank embeddings may be used to continue to prove new good results about algebraic structures which do not have proofs in ZFC. I therefore think it would be wise to search for a possible inconsistency of rank-into-rank cardinals so that when no inconsistency arises, plenty of algebraic results remain.
If there is an inconsistency arising from the algebras of elementary embeddings, then one can probably show that $n$-huge cardinals are inconsistent as well for fairly small $n$. On the other hand, the huge cardinals are probably safe from such an inconsistency.
I should mention that others in the set theory community have not expressed these doubts since the algebras with $\textrm{crit}((x*x)*y)>\textrm{crit}(x*y)$ are very new and no one else is working on them.
[1] Generalizations of Laver tables, Joseph Van Name (in progress; hopefully almost ready for Arxiv)