Medicineman, I'll give you the short version: you're wrong. Now for the long version:
A character with a skill of 6 in SR4 was at capped skill. That skill then becomes a 10, which is still a fair bit shy of the cap. But let's look at how these ranks are defined by their respective books:
The “best of the rest.” Maximum skill level for “rank-and-file” unnamed NPCs and starting characters.
You could easily sell your skills on the open market. This is the maximum skill level for starting player characters.
Completely different - one is just shy of being the absolute maximum anyone can acquire (rating 7 in SR4) and the other is midway through possible progression. Now let's look at what rank 10 says in SR5:
You are famous, even among the very best in your field.
Doesn't that sound almost the same as SR4's description of rank 6? Additionally, if you look at the NPCs that a skill-capped character will be facing in SR5, it's not unheard of for them to be facing enemies with a 10 or higher in at least one skill. Why would you want that character to go from being the "best of the rest" to inferior to the same enemies they faced in SR4?
Let's look at what I mean about NPC opposition. At skill-cap, it's probably not unheard of for a character to be facing Professional Rating 5 enemies. In SR4, the highest dice pool for a Professional Rating 5 enemy (Red Samurai) is 12 (Agility 7 + Firearms skill group 5). In SR5, that same enemy's highest dice pool is 16 (Agility 7 + Firearms skill group 9). That's a pretty gigantic difference of 4 dice. If you took a character who could deal with that Rating 5 enemy in SR4 and threw them right into SR5 against the same enemy they're going to either be slaughtered or at least have a hard time getting out alive.
Your suggestion that the karma values are the key factor in determining the value of the skill ignores all the other factors of the edition change. Namely that all the NPCs are substantially more powerful along with the characters. Because the damage codes are all different now too, there's no way to just take a SR4 character and move it into SR5 without doing some math.
OP, my best suggestion is to rebuild the character as closely as possible to the original intent of the build. Spend karma and nuyen using the SR5 rules. Your character will be a little weaker than before, but not substantially. And you'll have a great handle on some of the mechanics of SR5 that have changed from SR4. Limits are
not the only thing that need to be changed, despite Medicineman's protestations.