evaluation - How to compare power towers in Mathematica?

First I tried it directly, but it overflowed:

In[1]:= 2^3^2^3^5^8 < 3^2^2^7^6^7
During evaluation of In[1]:= General::ovfl: Overflow occurred in computation. >>
During evaluation of In[1]:= General::ovfl: Overflow occurred in computation. >>

Out[1]= Indeterminate < Indeterminate

Using double Log did not help either:

In[2]:= Log[Log[2]] + Log[3] 2^3^5^8 < Log[Log[3]] + Log[2] 2^7^6^7  
During evaluation of In[2]:= General::ovfl: Overflow occurred in computation. >>
During evaluation of In[2]:= General::ovfl: Overflow occurred in computation. >>

Out[2]= Overflow[] < Overflow[]

Then I tried to use the WolframAlpha function, and it seemed to be able to handle this and even much higher towers:

In[3]:= WolframAlpha["2^3^2^3^5^8 < 3^2^2^7^6^7", "MathematicaForms"][[2, 1]]
Out[3]= True


In[4]:= WolframAlpha["2^2^5^2^7^4^9^3^7^6^9^9^9^9^3^2 < 3^3^6^3^9^4^2^3^2^2^2^2^2^3^3^3", "MathematicaForms"][[2, 1]]
Out[4]= True

But then I discovered that it failed to perform certain comparisons:

In[5]:= WolframAlpha["4^6^8^8^9 < 3^2^3^3^3", "MathematicaForms"]
Out[5]= {}

And sometimes even gave wrong answers, even in seemingly trivial cases:

In[6]:= WolframAlpha["3^3^3^3^3 < 4^4^3^3^3", "MathematicaForms"][[2, 1]]
Out[6]= False

Can anybody suggest a correct and robust method to compare two expressions given in the form Hold[a^b^...^z], where a,b,...,z are positive integers?