= Solution
Regard the antisymmetric tensor as a real differential-form potential $A_p$. For a free <massless p-form gauge field>, its field strength is $F_{p+1}=dA_p$ and its gauge transformation is $A_p\mapsto A_p+d\Lambda_{p-1}$. The <Bianchi identity for an Abelian p-form> and the source-free field equation are
$$
dF=0,\qquad d*F=0.
$$
The <Hodge star> makes $*F$ a closed $(D-p-1)$-form. On a contractible patch, the <Poincaré lemma> gives $*F=dB_q$, hence
$$
\boxed{q_{\rm massless}=D-p-2,\qquad dB=*dA.}
$$
Conversely, the original Bianchi identity becomes the dual field equation $d*dB=0$, up to the harmless signature sign in $**$. The original equation becomes the dual Bianchi identity. This establishes <massless p-form duality> between gauge-equivalence classes of local solutions. It requires $0\le p\le D-2$ for an ordinary dual potential; global flux sectors and sources require extra data.
For a <massive p-form field> of mass $m>0$, let $\delta$ denote the formal <codifferential> and take its free equation to be
$$
\delta F+m^2A=0,\qquad F=dA.
$$
Applying $\delta$ yields $\delta A=0$, since $\delta^2=0$. Define
$$
B_q=\frac1m*F,\qquad\boxed{q_{\rm massive}=D-p-1.}
$$
Here the dual field is the Hodge dual of the field strength itself, rather than a potential for it. For definiteness, use one timelike direction and $**\omega_r=(-1)^{r(D-r)+1}\omega_r$. With $\delta\omega_r=(-1)^{D(r+1)}*d*\omega_r$, write $\eta=(-1)^{D(p+2)}$. The massive equation implies
$$
*dB=-\eta mA,\qquad A=-\frac\eta m*dB.
$$
This gives an invertible differential map at nonzero mass. Also $\delta B=0$ follows from $dF=0$, and substitution yields $\delta dB+m^2B=0$. Indeed the two codifferential signs multiply to $(-1)^{D(p+q+4)}=(-1)^{D(D+3)}=1$. Thus <massive p-form duality> exchanges the equations of two massive tensors of the displayed ranks.
The polarization counts confirm the distinction: a massive tensor in its rest frame has $\binom{D-1}{p}$ states, equal to $\binom{D-1}{D-p-1}$, whereas a massless gauge field has $\binom{D-2}{p}$, equal to $\binom{D-2}{D-p-2}$. The ordinary massive dual applies for $0\le p\le D-1$. A massless $(D-1)$-form has no local wave but may carry a constant top-form flux; its formal negative dual rank is a warning that it is not described by an ordinary dual potential. A degree-$D$ potential likewise has no local propagating field strength. These nondynamical endpoint cases do not change the duality of propagating fields.
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