= Solution
At a large separation, the two like net charges give the familiar leading repulsion $kQ_1Q_2/R^2$, where $k=(4\pi\epsilon_0)^{-1}$ and $R$ is centre separation. Each conductor also changes the other's surface charge distribution through <electrostatic induction>. The facing and remote surface charge densities need not remain equal, and an external insulating suspension prevents net charge transfer without preventing redistribution within either sphere.
\b[Like net charges do not guarantee repulsion for arbitrary conducting spheres at every separation.] Sufficiently unequal charges can give an attractive resultant after polarization. This does not contradict <Coulomb's law>: the law still applies to the distributed charges, while the point-charge approximation has lost the relevant induced contribution.
A useful controlled example takes a small conducting sphere of radius $a$ and fixed positive charge $Q_2$ in the nearly uniform field of a much more strongly charged distant sphere. For $a\ll R$, the small sphere has <electric polarizability> $\alpha=4\pi\epsilon_0a^3$ and induced dipole moment $p=\alpha E$. Its leading interaction energy is
$$
U(R)\simeq\frac{kQ_1Q_2}{R}-\frac12\alpha\left(\frac{kQ_1}{R^2}\right)^2.
$$
Differentiation gives the outward radial force
$$
\boxed{F_R\simeq\frac{kQ_1Q_2}{R^2}-\frac{2kQ_1^2a^3}{R^5}.}
$$
The second term is attractive. With $\eta=Q_2/Q_1>0$, this leading model gives $F_R<0$ when $\eta<2(a/R)^3$. One can choose a sufficiently small positive $\eta$ while retaining $a/R\ll1$, so the example does not require a neutral or earthed sphere. It is an asymptotic model, not an exact finite-sphere threshold.
\Image[/past-exam-of-the-medicine-course-of-the-university-of-oxford/1865/first-examination/physics-1865-like-charge-polarization.png]
{title=Repulsion and attraction in a leading small-sphere polarization model}
{description=Original calculated curves for fixed positive charge ratio Q2/Q1=0.001. The force is normalized by the outward point-charge force. The curve assumes a/R is small and retains the leading induced-dipole term; it is not an exact two-sphere calculation or experimental data.}
An exact analysis confirms that close-range attraction is possible for like-charged conducting spheres and identifies a special contact-charge ratio for which the force remains repulsive.<Electrostatics of Two Charged Conducting Spheres by John Lekner>[2012, Introduction and the like-charge attraction result]{ref} This later analysis supplies a precise qualification to the elementary expectation; the source question gives neither a charge ratio nor a separation regime that would justify asserting universal repulsion.
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