Strange behavior of old cone tweeters

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I came across some old no name "Model 22" speakers a few months ago. The cabinets are so well made, I just had to get them. The woofer is a CTS and the tweeter is a Quam-Nichols. They are early 1970's,

When I first listened to them, they were OK, except for a strong midrange peak. I identified this as the 1400 Hz tweeter resonance. The crossover is a 4.7 µf capacitor on the tweeter, which should be around 4000 Hz. The reason I'm hearing 1400 Hz is likely the impedance being very high at resonance. I already replaced the capacitors.

While I was playing around with them a bit, I noticed I can almost silence the resonance by putting a finger on the dust cap. I duplicated that with some tacky putty I had lying around. I don't have a calibrated mic but I can use a program to see differences in response. A small amount of putty, .1g causes a notch in the tweeter response with everything below the notch being lower. At a point around .7g, the notch becomes the roll-off you would expect. No amount of putty affects the response above the notch and touching the cone anywhere else doesn't tame that resonance. So here is what I find strange. I have always been taught that tweeters need to be very light. I also thought that that the center dome reproduced the highest frequencies from a driver.

So what should I do now. I could buy some 2nd order crossovers or add a choke but that might cost as much as replacing the tweeters. Since I may replace them anyway, should I try removing the dust cap to see what happens and possibly replace it with something else? Should I leave the putty on the domes? Should I try putting small holes in the dome to see if it's an air cavity resonance? I'm honestly surprised at the brightness of these tweeters when it isn't being buried under a nasty resonance.
 
My connection is bad right now but here is one picture.
 

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